Jensen's Inequality and its Role in Finance

Introduction

Jensen’s inequality is perhaps the most famous theorem in quantitative finance (note that it is a “theorem” and not a model or a formula) and it is the reason why financial derivatives have value. Concept of convexity, Jensen’s inequality, randomness and volatility of an asset price are intricately linked.

Definition

Jensen’s Inequality states that if is a convex function and is a random variable then

Example

You roll a die, and square the number of spots you get, finally you win that many dollars. For this exercise is , which is a convex function. So is divided by 6, so . But is , so is .

Conclusion

Jensen’s Inequality is a useful tool in mathematics, specifically in applied fields such as probability and statistics. Jensen’s Inequality and convexity can also be used to explain the relationship between randomness in stock prices and the value inherent in options, the latter typically having some convexity. A common application of the inequality is in the comparison of arithmetic and geometric means when averaging the financial returns for a time interval.

References

1. https://ebrary.net/7091/business_finance/what_jensens_inequality_and_what_its_role_finance
2. https://www.risklatte.xyz/Articles/QuantitativeFinance/QF187.php
3. Wilmott, Paul Wilmott on Quantitative Finance (2006)
4. https://machinelearningmastery.com/a-gentle-introduction-to-jensens-inequality/